16.1 Vector Fields

Claudia Castro-Castro
Math 283 Spring 2020

Instructions:

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Outline

  • The following topics will be covered in this section:
    • Definition of vector fiels in 2-D
    • Vector plots
    • Definition of vector fiels in 3-D
    • Streamlines
    • Gradient fields
    • Conservative vector fields

Recall:

  • Calculus I&II

    • Scalar functions of one variable

      \[ f:\mathbb{R} \rightarrow \mathbb{R} \]
  • Calculus III

    • Vector functions of one variable

      \[ \vec{r}:\mathbb{R} \rightarrow \mathbb{R}^n \]

    • Scalar functions of several variables

      \[ f:\mathbb{R}^n \rightarrow \mathbb{R} \]

    • Vector fields of several variables

      \[ \mathbf{\vec{F}}:\mathbb{R}^n \rightarrow \mathbb{R}^n \]

Definition of Vector Field

  • In general, a vector field is a function whose domain is a set of points in \( \mathbb{R}^n \) and whose range is a set of vectors in \( V_n \)
  • If \( n=2 \)

    DEFINITION Let \( D \) be a set in \( \mathbb{R}^2 \) (a plane region). A vector field on \( \mathbb{R}^2 \) is a function \( \mathbf{\vec{F}} \) that assigns to each point \( (x,y) \) in \( D \) a two-dimensional vector \( \mathbf{\vec{F}}(x,y) \)

Vector plot 2-D Vector Field

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • In terms of its component functions \( P \) and \( Q \) \[ \begin{align} \mathbf{\vec{F}}(x,y)&=P(x,y)\mathbf{\hat{i}}+Q(x,y)\mathbf{\hat{j}}\\ &= \langle P(x,y),\;Q(x,y)\rangle \end{align} \]
  • or \[ \begin{align} \mathbf{\vec{F}}&=P\mathbf{\hat{i}}+Q\mathbf{\hat{j}}\\ &=\langle P,\;Q\rangle \end{align} \]

Example 1

  • Sketch a vector plot for the vector field \[ \mathbf{\hat{F}}(x,y)=\langle 1,\;-1\rangle \]
2-D Grided plane

  • Table inputs vs outputs
    \( (x,y) \) \( \mathbf{\vec{F}}(x,y) \)
    \( (0,0) \) \( \langle 1,\;-1\rangle \)

Example 1 cont'd

  • Sketch a vector plot for the vector field \[ \mathbf{\hat{F}}(x,y)=\langle 1,\;-1\rangle \]
2-D constant Vector plot

  • Table inputs vs outputs
    \( (x,y) \) \( \mathbf{\vec{F}}(x,y) \)
    \( (0,0) \) \( \langle 1,\;-1\rangle \)
    \( (1,0) \) \( \langle 1,\;-1\rangle \)
    \( (2,0) \) \( \langle 1,\;-1\rangle \)
    \( (0,1) \) \( \langle 1,\;-1\rangle \)
    \( (1,1) \) \( \langle 1,\;-1\rangle \)
    \( (2,1) \) \( \langle 1,\;-1\rangle \)
    \( \vdots \) \( \vdots \)

Example 2

  • Sketch a vector plot for the vector field \[ \mathbf{\hat{F}}(x,y)=\langle -y,\;x\rangle \]
2-D Grided plane

  • Table inputs vs outputs
    \( (x,y) \) \( \mathbf{\vec{F}}(x,y)=\langle -y,x\rangle \)

Example 2

  • Sketch a vector plot for the vector field \[ \mathbf{\hat{F}}(x,y)=\langle -y,\;x\rangle \]
2-D Grided plane

  • Table inputs vs outputs
    \( (x,y) \) \( \mathbf{\vec{F}}(x,y)=\langle -y,x\rangle \)
    \( (0,0) \) \( \langle 0,\;0\rangle \)
    \( (1,0) \) \( \langle -0,\;1\rangle \)
    \( (1,1) \) \( \langle -1,\;1\rangle \)
    \( (0,1) \) \( \langle -1,\;0\rangle \)
    \( (0,2) \) \( \langle -2,\;0\rangle \)
    \( (-1,1) \) \( \langle -1,\;-1\rangle \)
    \( (-1,0) \) \( \langle 0,\;-1\rangle \)
    \( (-1,-1) \) \( \langle 1,\;-1\rangle \)
    \( (0,-1) \) \( \langle 1,\;0\rangle \)
    \( (1,-1) \) \( \langle 1,\;1\rangle \)

Example 2 cont'd

  • Vector plot for the vector field \[ \mathbf{\hat{F}}(x,y)=\langle -y,\;x\rangle \]
Computer generated vector plot 2-D Grided plane

Wolfram Cloud Notebook

Other examples

Source vector field

\[ \mathbf{\hat{F}}(x,y)=\langle x,\;y\rangle \]

2-d vector field plot

\[ \mathbf{\hat{F}}(x,y)=\langle y,\;\sin{x}\rangle \]

2-d magnetic field

\[ \mathbf{\hat{F}}(x,y)=\langle x^2-y^2,\;2xy\rangle \]

2-d gravitational field

\[ \mathbf{\hat{F}}(x,y)=mMG\frac{\langle -x,-y\rangle}{(x^2+y^2)^{\frac{3}{2}}} \]

Velocity vector fields

  • The speed at any given point is indicated by the length of the arrow.
San Francisco wind patterns and Ocean currents off of Nova Scotia

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

3-D Vector Field

  • DEFINITION Let \( E \) be a set in \( \mathbb{R}^3 \). A vector field on \( \mathbb{R}^3 \) is a function \( \mathbf{\vec{F}} \) that assigns to each point \( (x,y,z) \) in \( E \) a three-dimensional vector \( \mathbf{\vec{F}}(x,y,z) \)

Vector plot 3-D Vector Field

Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition

  • In terms of its component functions \( P \), \( Q \), and \( R \) \[ \begin{align} \mathbf{\vec{F}}(x,y,z)&=P(x,y,z)\mathbf{\hat{i}}+Q(x,y,z)\mathbf{\hat{j}}+R(x,y,z)\mathbf{\hat{k}}\\ &= \langle P(x,y),\;Q(x,y)\rangle \end{align} \]
  • or \[ \begin{align} \mathbf{\vec{F}}&=P\mathbf{\hat{i}}+Q\mathbf{\hat{j}}+R\mathbf{\hat{k}}\\ &=\langle P,\;Q,\;R\rangle \end{align} \]

3D Examples

Constant vector field Constant 3-d vector field

\[ \mathbf{\hat{F}}(x,y,z)=\langle 1,\;1,-1\rangle \]

Gravitational field Constant 3-d vector field

\[ \mathbf{\hat{F}}(x,y,z)=mMG\frac{\langle -x,\;-y,-z\rangle}{(x^2+y^2+z^2)^{\frac{3}{2}}} \]

Velocity field in fluid flow Velocity field in fluid flow

Courtesy of J. Stewart, Calculus: Early transcendentals, 2nd edition

Streamlines

  • The flow lines or streamlines of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus thevectors in a vector field are tangent to the flow lines
  • Flow lines are curves \( \vec{r}(t)= \langle x(t),\;y(t)\rangle \) such that \[ \frac{dx}{dt}=P(x,y) \quad\quad \frac{dy}{dt}=Q(x,y) \]
  • Previous examples now with streamlines
Rotational field with streamlines

2-d field with streamlines

Gradient Fields

  • Recall: If \( f \) is a scalar function of two variables, we know that its gradient \( \nabla f \) is \[ \nabla f = \frac{\partial}{\partial x} f(x,y) \mathbf{\hat{i}} + \frac{\partial}{\partial y} (x,y) \mathbf{\hat{j}}= \langle \frac{\partial}{\partial x} f(x,y), \frac{\partial}{\partial y} (x,y)\rangle \]
  • \( \nabla f \) is really a vector field on \( \mathbb{R}^2 \) and is called a gradient vector field

Example

  • Consider the scalar field \( f(x,y)=x^2y-y^3 \).
    (a) Find the gradient vector field \( \mathbf{\hat{F}}=\nabla f \).
    (b) Plot the gradient vector field \( \mathbf{\hat{F}} \) together with a contour plot of \( f \)
    • (a) The gradient vector field is \[ \mathbf{\hat{F}}(x,y)= \frac{\partial}{\partial x} f(x,y) \mathbf{\hat{i}} + \frac{\partial}{\partial y} f(x,y) \mathbf{\hat{j}} \]
    • \[ \mathbf{\hat{F}}(x,y)= \frac{\partial}{\partial x}\left(x^2y-y^3 \right) \mathbf{\hat{i}} + \frac{\partial}{\partial y} \left(x^2y-y^3 \right) \mathbf{\hat{j}} \]
    • \[ \begin{align} \mathbf{\hat{F}}(x,y)&= 2xy\; \mathbf{\hat{i}} + \left(x^2-3y^2 \right) \mathbf{\hat{j}} \\ & = \langle 2xy,\; x^2-3y^2 \rangle \end{align} \]

Gradient Fields

  • (b) Plot the gradient vector field \( \mathbf{\hat{F}}= \langle 2xy,\; x^2-3y^2 \rangle \) together with a contour plot of \( f(x,y)=x^2y-y^3 \)
Gradiend field plus contours of potential function

  • \( \star \) The gradient vectors are perpendicular to the level curves.
  • \( \star \) The gradient vectors are long where the level curves are close to each other and short where they are farther apart.
  • \( \star \) The length of the gradient vector is the value of the directional derivative of \( f \) and closely spaced level curves indicate a steep graph.

Conservative fields

  • A vector field \( \mathbf{\vec{F}} \) is called a conservative vector field if it is the gradient of some scalar function, that is, if there exists a function \( f \) such that \( \mathbf{\vec{F}} \) = \( \nabla f \). In this situation \( f \) is called a potential function for \( \mathbf{\vec{F}} \)
  • Conservative fields arise frequently in physics

Example: Gravitational field

  • Consider the scalar function \[ f(x,y,z)=\frac{mMG}{\sqrt{x^2+y^2+z^2}} \] where \( m \) and \( M \) are the masses of two bodies, and \( G \) is the gravitational constant
  • Its gradient field is \[ \mathbf{\hat{F}}(x,y)= \frac{\partial f}{\partial x} \mathbf{\hat{i}} + \frac{\partial f }{\partial y} \mathbf{\hat{j}}+ \frac{\partial f }{\partial z} \mathbf{\hat{k}} \]
  • \[ \mathbf{\hat{F}}(x,y,z)=mMG \frac{-x}{(x^2+y^2+z^2)^{\frac{3}{2}}} \mathbf{\hat{i}} + mMG \frac{-y}{(x^2+y^2+z^2)^{\frac{3}{2}}} \mathbf{\hat{j}} + mMG \frac{-z}{(x^2+y^2+z^2)^{\frac{3}{2}}} \mathbf{\hat{k}} \]
Constant 3-d vector field

Final remarks

  • A vector field assigns a vector to each point in its domain
  • We can sketch vector plots by determining the magnitude and direction of a vector in various locations in the domain.
  • A vector field is called conservative if \( \mathbf{\vec{F}}=\nabla f \)
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  • Questions